10 21: Difference between revisions

Revision as of 19:06, 28 August 2005

10_20

10_22

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10 21 Quick Notes

10 21 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_5,14,6,15 X_3,13,4,12 X_13,3,14,2 X_7,16,8,17 X_11,20,12,1 X_15,6,16,7 X_19,8,20,9 X_9,18,10,19 X_17,10,18,11
Gauss code	-1, 4, -3, 1, -2, 7, -5, 8, -9, 10, -6, 3, -4, 2, -7, 5, -10, 9, -8, 6
Dowker-Thistlethwaite code	4 12 14 16 18 20 2 6 10 8
Conway Notation	[3412]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-13][1]
Hyperbolic Volume	9.67514
A-Polynomial	See Data:10 21/A-polynomial

[edit Notes for 10 21's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$2$
Topological 4 genus	$2$
Concordance genus	$2$
Rasmussen s-Invariant	-4

[edit Notes for 10 21's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+7t^{2}-9t+9-9t^{-1}+7t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-5z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 45, -4 }
Jones polynomial	$1-2q^{-1}+4q^{-2}-5q^{-3}+7q^{-4}-7q^{-5}+7q^{-6}-6q^{-7}+3q^{-8}-2q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{8}+3z^{2}a^{8}+a^{8}-z^{6}a^{6}-4z^{4}a^{6}-5z^{2}a^{6}-3a^{6}-z^{6}a^{4}-3z^{4}a^{4}+2a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-2z^{2}a^{12}+2z^{5}a^{11}-4z^{3}a^{11}+2za^{11}+2z^{6}a^{10}-2z^{4}a^{10}+2z^{7}a^{9}-2z^{5}a^{9}+3za^{9}+2z^{8}a^{8}-5z^{6}a^{8}+9z^{4}a^{8}-5z^{2}a^{8}+a^{8}+z^{9}a^{7}-z^{7}a^{7}+2z^{3}a^{7}-za^{7}+4z^{8}a^{6}-14z^{6}a^{6}+20z^{4}a^{6}-14z^{2}a^{6}+3a^{6}+z^{9}a^{5}-z^{7}a^{5}-3z^{5}a^{5}+3z^{3}a^{5}-2za^{5}+2z^{8}a^{4}-6z^{6}a^{4}+4z^{4}a^{4}-3z^{2}a^{4}+2a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-4z^{4}a^{2}+4z^{2}a^{2}-a^{2}$
The A2 invariant	$q^{30}-2q^{22}-2q^{18}+q^{14}+3q^{10}+q^{6}+1$
The G2 invariant	$q^{162}-q^{160}+2q^{158}-3q^{156}+q^{154}-q^{152}-3q^{150}+6q^{148}-7q^{146}+7q^{144}-6q^{142}+3q^{140}+4q^{138}-9q^{136}+12q^{134}-14q^{132}+12q^{130}-7q^{128}+q^{126}+6q^{124}-12q^{122}+24q^{120}-18q^{118}+14q^{116}-7q^{114}-4q^{112}+15q^{110}-20q^{108}+17q^{106}-9q^{104}-3q^{102}+12q^{100}-15q^{98}+5q^{96}+6q^{94}-20q^{92}+21q^{90}-20q^{88}+2q^{86}+16q^{84}-33q^{82}+38q^{80}-32q^{78}+13q^{76}+8q^{74}-28q^{72}+36q^{70}-34q^{68}+22q^{66}-4q^{64}-13q^{62}+25q^{60}-21q^{58}+13q^{56}+4q^{54}-15q^{52}+19q^{50}-13q^{48}-q^{46}+19q^{44}-28q^{42}+30q^{40}-16q^{38}-2q^{36}+21q^{34}-29q^{32}+30q^{30}-21q^{28}+7q^{26}+6q^{24}-16q^{22}+18q^{20}-13q^{18}+9q^{16}-q^{14}-2q^{12}+4q^{10}-4q^{8}+3q^{6}-q^{4}+q^{2}$

Further Quantum Invariants

Further quantum knot invariants for 10_21.

A1 Invariants.

Weight	Invariant
1	$q^{21}-q^{19}+q^{17}-3q^{15}+q^{13}+2q^{7}-q^{5}+2q^{3}-q+q^{-1}$
2	$q^{58}-q^{56}-q^{54}+2q^{52}-2q^{50}-q^{48}+4q^{46}-4q^{44}-q^{42}+9q^{40}-5q^{38}-3q^{36}+7q^{34}-3q^{32}-5q^{30}+q^{28}+3q^{26}-3q^{24}-3q^{22}+6q^{20}-7q^{16}+6q^{14}+4q^{12}-7q^{10}+3q^{8}+6q^{6}-5q^{4}-q^{2}+4-q^{-2}-q^{-4}+q^{-6}$
3	$q^{111}-q^{109}-q^{107}+2q^{103}-2q^{99}-q^{97}+q^{95}+2q^{93}+q^{89}-3q^{87}-2q^{85}+4q^{83}+9q^{81}-6q^{79}-14q^{77}-q^{75}+21q^{73}+2q^{71}-24q^{69}-6q^{67}+20q^{65}+16q^{63}-15q^{61}-14q^{59}+5q^{57}+17q^{55}+2q^{53}-13q^{51}-12q^{49}+12q^{47}+15q^{45}-11q^{43}-21q^{41}+7q^{39}+22q^{37}-3q^{35}-23q^{33}-3q^{31}+23q^{29}+9q^{27}-18q^{25}-16q^{23}+13q^{21}+20q^{19}-4q^{17}-19q^{15}-3q^{13}+17q^{11}+10q^{9}-11q^{7}-10q^{5}+4q^{3}+10q-6q^{-3}-q^{-5}+3q^{-7}+q^{-9}-q^{-11}-q^{-13}+q^{-15}$
4	$q^{180}-q^{178}-q^{176}+4q^{170}-2q^{168}-2q^{166}-2q^{164}-3q^{162}+10q^{160}+2q^{158}-q^{156}-7q^{154}-12q^{152}+13q^{150}+10q^{148}+9q^{146}-10q^{144}-31q^{142}+2q^{140}+17q^{138}+36q^{136}+5q^{134}-55q^{132}-34q^{130}+6q^{128}+75q^{126}+54q^{124}-56q^{122}-79q^{120}-44q^{118}+83q^{116}+113q^{114}-12q^{112}-94q^{110}-101q^{108}+34q^{106}+115q^{104}+45q^{102}-41q^{100}-98q^{98}-28q^{96}+58q^{94}+61q^{92}+22q^{90}-48q^{88}-54q^{86}-8q^{84}+45q^{82}+57q^{80}-5q^{78}-63q^{76}-47q^{74}+36q^{72}+75q^{70}+23q^{68}-73q^{66}-78q^{64}+30q^{62}+89q^{60}+51q^{58}-69q^{56}-102q^{54}+3q^{52}+79q^{50}+86q^{48}-27q^{46}-100q^{44}-42q^{42}+29q^{40}+93q^{38}+32q^{36}-50q^{34}-58q^{32}-38q^{30}+50q^{28}+57q^{26}+16q^{24}-21q^{22}-62q^{20}-9q^{18}+25q^{16}+38q^{14}+25q^{12}-33q^{10}-25q^{8}-12q^{6}+14q^{4}+30q^{2}-1-7q^{-2}-15q^{-4}-4q^{-6}+12q^{-8}+2q^{-10}+2q^{-12}-4q^{-14}-3q^{-16}+3q^{-18}+q^{-22}-q^{-24}-q^{-26}+q^{-28}$
5	$q^{265}-q^{263}-q^{261}+2q^{255}+2q^{253}-2q^{251}-4q^{249}-q^{247}+5q^{243}+8q^{241}-9q^{237}-9q^{235}-4q^{233}+8q^{231}+18q^{229}+10q^{227}-12q^{225}-25q^{223}-17q^{221}+9q^{219}+31q^{217}+33q^{215}-3q^{213}-50q^{211}-51q^{209}-5q^{207}+59q^{205}+89q^{203}+36q^{201}-78q^{199}-139q^{197}-79q^{195}+77q^{193}+209q^{191}+162q^{189}-63q^{187}-273q^{185}-273q^{183}-5q^{181}+327q^{179}+402q^{177}+96q^{175}-324q^{173}-507q^{171}-250q^{169}+271q^{167}+574q^{165}+384q^{163}-155q^{161}-554q^{159}-477q^{157}-q^{155}+466q^{153}+518q^{151}+126q^{149}-322q^{147}-461q^{145}-231q^{143}+152q^{141}+378q^{139}+265q^{137}-22q^{135}-241q^{133}-264q^{131}-91q^{129}+138q^{127}+235q^{125}+146q^{123}-45q^{121}-209q^{119}-193q^{117}+q^{115}+201q^{113}+226q^{111}+34q^{109}-218q^{107}-266q^{105}-52q^{103}+238q^{101}+320q^{99}+86q^{97}-261q^{95}-383q^{93}-135q^{91}+264q^{89}+437q^{87}+213q^{85}-230q^{83}-477q^{81}-299q^{79}+155q^{77}+474q^{75}+376q^{73}-40q^{71}-421q^{69}-439q^{67}-87q^{65}+316q^{63}+438q^{61}+207q^{59}-167q^{57}-386q^{55}-285q^{53}+18q^{51}+273q^{49}+297q^{47}+108q^{45}-128q^{43}-246q^{41}-178q^{39}-4q^{37}+148q^{35}+177q^{33}+94q^{31}-31q^{29}-124q^{27}-128q^{25}-49q^{23}+49q^{21}+101q^{19}+91q^{17}+25q^{15}-56q^{13}-84q^{11}-55q^{9}+4q^{7}+50q^{5}+60q^{3}+27q-18q^{-1}-40q^{-3}-32q^{-5}-3q^{-7}+17q^{-9}+23q^{-11}+13q^{-13}-6q^{-15}-12q^{-17}-7q^{-19}+2q^{-23}+5q^{-25}+2q^{-27}-3q^{-29}-q^{-31}+q^{-33}+q^{-39}-q^{-41}-q^{-43}+q^{-45}$

A2 Invariants.

Weight	Invariant
1,0	$q^{30}-2q^{22}-2q^{18}+q^{14}+3q^{10}+q^{6}+1$
1,1	$q^{84}-2q^{82}+4q^{80}-8q^{78}+13q^{76}-18q^{74}+22q^{72}-30q^{70}+37q^{68}-40q^{66}+42q^{64}-48q^{62}+57q^{60}-54q^{58}+52q^{56}-56q^{54}+47q^{52}-30q^{50}+8q^{48}+24q^{46}-56q^{44}+96q^{42}-128q^{40}+154q^{38}-174q^{36}+170q^{34}-160q^{32}+132q^{30}-102q^{28}+54q^{26}-6q^{24}-32q^{22}+67q^{20}-90q^{18}+106q^{16}-98q^{14}+88q^{12}-72q^{10}+56q^{8}-34q^{6}+23q^{4}-12q^{2}+6-2q^{-2}+q^{-4}$
2,0	$q^{76}-q^{70}-q^{64}-q^{62}-2q^{60}+2q^{56}+3q^{54}-2q^{52}+3q^{50}+6q^{48}+2q^{46}-5q^{44}-3q^{42}+2q^{40}-4q^{38}-5q^{36}+q^{32}-3q^{30}+q^{28}+q^{26}-q^{24}+q^{22}+5q^{20}+q^{18}-3q^{16}+2q^{14}+5q^{12}-q^{10}-3q^{8}+2q^{6}+3q^{4}-1+q^{-4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{68}-q^{66}+q^{62}-4q^{60}-q^{58}+4q^{56}-3q^{54}-q^{52}+10q^{50}-q^{48}-3q^{46}+6q^{44}-2q^{42}-5q^{40}+q^{38}-3q^{34}-4q^{32}+q^{30}-7q^{26}+3q^{24}+5q^{22}-4q^{20}+3q^{18}+6q^{16}-3q^{14}+3q^{12}+3q^{10}-2q^{8}+2q^{6}+q^{4}-q^{2}+1$
1,0,0	$q^{39}+q^{35}-q^{33}+q^{31}-2q^{29}-3q^{25}-q^{23}-q^{21}+2q^{17}+q^{15}+3q^{13}+2q^{9}-q^{7}+q^{5}+q$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{86}-q^{82}+q^{80}+q^{78}-4q^{76}-4q^{74}+q^{72}-4q^{68}+9q^{64}+5q^{62}+q^{60}+7q^{58}+6q^{56}-3q^{54}-2q^{52}-6q^{48}-8q^{46}-2q^{44}-3q^{42}-9q^{40}-4q^{38}+4q^{36}-q^{34}-4q^{32}+5q^{30}+5q^{28}+q^{26}+q^{24}+4q^{22}+3q^{20}+2q^{18}+2q^{16}+2q^{14}+q^{12}+q^{10}+q^{8}+q^{2}$
1,0,0,0	$q^{48}+q^{44}+q^{38}-2q^{36}-3q^{32}-2q^{30}-2q^{28}-q^{26}+q^{22}+3q^{20}+q^{18}+3q^{16}+2q^{12}+q^{6}+q^{2}$

B2 Invariants.

Weight

Invariant

0,1

q^{68}-q^{66}+2q^{64}-3q^{62}+4q^{60}-5q^{58}+6q^{56}-7q^{54}+7q^{52}-8q^{50}+5q^{48}-3q^{46}+4q^{42}-7q^{40}+11q^{38}-14q^{36}+15q^{34}-16q^{32}+13q^{30}-12q^{28}+9q^{26}-5q^{24}+q^{22}+4q^{20}-5q^{18}+8q^{16}-7q^{14}+9q^{12}-7q^{10}+6q^{8}-4q^{6}+3q^{4}-q^{2}+1

1,0

q^{110}-q^{106}-q^{104}+q^{102}+2q^{100}-q^{98}-4q^{96}-3q^{94}+2q^{92}+5q^{90}+q^{88}-5q^{86}-4q^{84}+4q^{82}+10q^{80}+2q^{78}-6q^{76}-4q^{74}+5q^{72}+6q^{70}-3q^{68}-7q^{66}-q^{64}+5q^{62}+q^{60}-6q^{58}-4q^{56}+3q^{54}+3q^{52}-4q^{50}-4q^{48}+2q^{46}+4q^{44}-2q^{42}-7q^{40}+8q^{36}+5q^{34}-6q^{32}-7q^{30}+3q^{28}+10q^{26}+3q^{24}-6q^{22}-5q^{20}+5q^{18}+6q^{16}-4q^{12}-q^{10}+3q^{8}+2q^{6}-q^{4}-q^{2}+q^{-2}

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{94}-q^{92}+q^{90}-2 q^{88}+2 q^{86}-4 q^{84}+2 q^{82}-4 q^{80}+5 q^{78}-5 q^{76}+5 q^{74}-4 q^{72}+9 q^{70}-3 q^{68}+4 q^{66}-2 q^{64}+2 q^{62}+2 q^{60}-4 q^{58}+4 q^{56}-8 q^{54}+9 q^{52}-11 q^{50}+10 q^{48}-15 q^{46}+9 q^{44}-13 q^{42}+7 q^{40}-11 q^{38}+5 q^{36}-4 q^{34}+3 q^{32}+2 q^{30}+7 q^{26}-3 q^{24}+7 q^{22}-5 q^{20}+9 q^{18}-5 q^{16}+6 q^{14}-4 q^{12}+5 q^{10}-2 q^8+2 q^6-q^4+q^2}

G2 Invariants.

Weight

Invariant

1,0

q^{162}-q^{160}+2q^{158}-3q^{156}+q^{154}-q^{152}-3q^{150}+6q^{148}-7q^{146}+7q^{144}-6q^{142}+3q^{140}+4q^{138}-9q^{136}+12q^{134}-14q^{132}+12q^{130}-7q^{128}+q^{126}+6q^{124}-12q^{122}+24q^{120}-18q^{118}+14q^{116}-7q^{114}-4q^{112}+15q^{110}-20q^{108}+17q^{106}-9q^{104}-3q^{102}+12q^{100}-15q^{98}+5q^{96}+6q^{94}-20q^{92}+21q^{90}-20q^{88}+2q^{86}+16q^{84}-33q^{82}+38q^{80}-32q^{78}+13q^{76}+8q^{74}-28q^{72}+36q^{70}-34q^{68}+22q^{66}-4q^{64}-13q^{62}+25q^{60}-21q^{58}+13q^{56}+4q^{54}-15q^{52}+19q^{50}-13q^{48}-q^{46}+19q^{44}-28q^{42}+30q^{40}-16q^{38}-2q^{36}+21q^{34}-29q^{32}+30q^{30}-21q^{28}+7q^{26}+6q^{24}-16q^{22}+18q^{20}-13q^{18}+9q^{16}-q^{14}-2q^{12}+4q^{10}-4q^{8}+3q^{6}-q^{4}+q^{2}

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 21"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= $-2t^{3}+7t^{2}-9t+9-9t^{-1}+7t^{-2}-2t^{-3}$

In[5]:= Conway[K][z]

Out[5]= $-2z^{6}-5z^{4}+z^{2}+1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= $\{1\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 45, -4 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= $1-2q^{-1}+4q^{-2}-5q^{-3}+7q^{-4}-7q^{-5}+7q^{-6}-6q^{-7}+3q^{-8}-2q^{-9}+q^{-10}$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $z^{4}a^{8}+3z^{2}a^{8}+a^{8}-z^{6}a^{6}-4z^{4}a^{6}-5z^{2}a^{6}-3a^{6}-z^{6}a^{4}-3z^{4}a^{4}+2a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}$

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= $z^{4}a^{12}-2z^{2}a^{12}+2z^{5}a^{11}-4z^{3}a^{11}+2za^{11}+2z^{6}a^{10}-2z^{4}a^{10}+2z^{7}a^{9}-2z^{5}a^{9}+3za^{9}+2z^{8}a^{8}-5z^{6}a^{8}+9z^{4}a^{8}-5z^{2}a^{8}+a^{8}+z^{9}a^{7}-z^{7}a^{7}+2z^{3}a^{7}-za^{7}+4z^{8}a^{6}-14z^{6}a^{6}+20z^{4}a^{6}-14z^{2}a^{6}+3a^{6}+z^{9}a^{5}-z^{7}a^{5}-3z^{5}a^{5}+3z^{3}a^{5}-2za^{5}+2z^{8}a^{4}-6z^{6}a^{4}+4z^{4}a^{4}-3z^{2}a^{4}+2a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-4z^{4}a^{2}+4z^{2}a^{2}-a^{2}$

Vassiliev invariants

V₂ and V₃:

(1, 0)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

4

0

8

{\frac {62}{3}}

{\frac {130}{3}}

0

-224

-96

-224

{\frac {32}{3}}

0

{\frac {248}{3}}

{\frac {520}{3}}

{\frac {33391}{30}}

-{\frac {4234}{5}}

{\frac {91742}{45}}

{\frac {3665}{18}}

{\frac {8431}{30}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j-2r=s+1

or

j-2r=s-1

, where

s=

-4 is the signature of 10 21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

1

-1

-3

3

1

2

-5

3

2

-1

-7

4

2

-9

3

0

-11

4

0

-13

2

3

1

-15

1

4

-3

-17

1

2

1

-19

1

-1

-21

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 21]]
Out[2]=	10
In[3]:=	PD[Knot[10, 21]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 14, 6, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], X[7, 16, 8, 17], X[11, 20, 12, 1], X[15, 6, 16, 7], X[19, 8, 20, 9], X[9, 18, 10, 19], X[17, 10, 18, 11]]
In[4]:=	GaussCode[Knot[10, 21]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 7, -5, 8, -9, 10, -6, 3, -4, 2, -7, 5, -10, 9, -8, 6]
In[5]:=	BR[Knot[10, 21]]
Out[5]=	BR[4, {-1, -1, -2, 1, -2, -2, -2, -2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[10, 21]][t]
Out[6]=	2 7 9 2 3 9 - -- + -- - - - 9 t + 7 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 21]][z]
Out[7]=	2 4 6 1 + z - 5 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 21], Knot[11, NonAlternating, 69]}
In[9]:=	{KnotDet[Knot[10, 21]], KnotSignature[Knot[10, 21]]}
Out[9]=	{45, -4}
In[10]:=	J=Jones[Knot[10, 21]][q]
Out[10]=	-10 2 3 6 7 7 7 5 4 2 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 21]}
In[12]:=	A2Invariant[Knot[10, 21]][q]
Out[12]=	-30 2 2 -14 3 -6 1 + q - --- - --- + q + --- + q 22 18 10 q q q
In[13]:=	Kauffman[Knot[10, 21]][a, z]
Out[13]=	2 4 6 8 5 7 9 11 2 2 -a + 2 a + 3 a + a - 2 a z - a z + 3 a z + 2 a z + 4 a z - 4 2 6 2 8 2 12 2 3 3 5 3 3 a z - 14 a z - 5 a z - 2 a z + 5 a z + 3 a z + 7 3 11 3 2 4 4 4 6 4 8 4 2 a z - 4 a z - 4 a z + 4 a z + 20 a z + 9 a z - 10 4 12 4 3 5 5 5 9 5 11 5 2 6 2 a z + a z - 7 a z - 3 a z - 2 a z + 2 a z + a z - 4 6 6 6 8 6 10 6 3 7 5 7 7 7 6 a z - 14 a z - 5 a z + 2 a z + 2 a z - a z - a z + 9 7 4 8 6 8 8 8 5 9 7 9 2 a z + 2 a z + 4 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 21]], Vassiliev[3][Knot[10, 21]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 21]][q, t]
Out[15]=	2 3 1 1 1 2 1 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 q q q t q t q t q t q t q t 2 3 4 4 3 3 4 2 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t 3 t t 2 ---- + -- + - + q t 5 3 q q t q

@@ Line 1: / Line 1: @@
 <!--  -->
+<!-- -->
+<!--  -->
+<!-- -->
 <!-- provide an anchor so we can return to the top of the page -->
 <span id="top"></span>
+<!-- -->
 <!-- this relies on transclusion for next and previous links -->
 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=10|k=21|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-2,7,-5,8,-9,10,-6,3,-4,2,-7,5,-10,9,-8,6/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=10|k=21|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-2,7,-5,8,-9,10,-6,3,-4,2,-7,5,-10,9,-8,6/goTop.html}}
-|{{:{{PAGENAME}} Quick Notes}}
-|}
 <br style="clear:both" />
@@ Line 24: / Line 21: @@
 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=13.3333%><table cellpadding=0 cellspacing=0>
@@ Line 48: / Line 41: @@
 <tr align=center><td>-19</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
 <tr align=center><td>-21</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 133: / Line 125: @@
   q  t   q</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

10 21: Difference between revisions

Revision as of 19:06, 28 August 2005

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